If $X$ and $Y$ are two sets such that $n(X) = 17$,$n(Y) = 23$ and $n(X \cup Y) = 38$,find $n(X \cap Y)$.

If ${A_1}, {A_2}, {A_3}, \dots, {A_{30}}$ are $30$ sets each having $5$ elements and ${B_1}, {B_2}, \dots, {B_n}$ are $n$ sets each having $3$ elements. Let $\bigcup_{i=1}^{30} {A_i} = \bigcup_{j=1}^n {B_j} = S$ and each element of $S$ belongs to exactly $10$ of the $A_i$'s and exactly $9$ of the $B_j$'s,then $n$ is equal to:

Let $A = \{1, 2, 3, 4, 5, 6, 7\}$ and $B = \{3, 6, 7, 9\}$. Then the number of elements in the set $\{ C \subseteq A : C \cap B \neq \phi \}$ is

The number of $4$-digit numbers which are neither a multiple of $7$ nor a multiple of $3$ is ..... .

In a class of $55$ students,$23$ study Mathematics,$24$ study Physics,$19$ study Chemistry,$12$ study Mathematics and Physics,$9$ study Mathematics and Chemistry,$7$ study Physics and Chemistry,and $4$ study all three subjects. Find the number of students who study exactly one subject.

Two dice are thrown. The events $A, B$ and $C$ are as follows:
$A:$ getting an even number on the first die.
$B:$ getting an odd number on the first die.
$C:$ getting the sum of the numbers on the dice $\leq 5$.
Describe the event $A$ but not $C$.